\((\sqrt[5]{8})^{\frac{5}{2}} \times(16)^{\frac{-3}{2}}\)
\(=\left[\left(2^{3}\right)^{\frac{1}{5}}\right]^{\frac{5}{2}} \times\left(2^{4}\right)^{\frac{-3}{2}}\)
\(=2^{\frac{3}{5}} \times \frac{5}{2} \times 2^{4} \times\left(-\frac{3}{2}\right)\)
\(=2^{\frac{3}{2}} \times 2^{-6}\)
\(=2^{\frac{3}{2}-6}\)
\(=2^{\frac{3-12}{2}}\)
\(=2^{\frac{-9}{2}}\)

\(\left\{(125)^{-2} \times(16)^{\frac{-3}{2}}\right\}^{\frac{-1}{6}}\)
\(=\left\{\left(5^{3}\right)^{-2} \times\left(2^{4}\right)^{\frac{-3}{2}}\right\}^{\frac{-1}{6}}\)
\(=\left\{5^{-6} \times 2^{-6}\right\}^{\frac{-1}{6}}\)
\(=(5 \times 2)^{(-6) \times\left(\frac{1}{-6}\right)}\)
\(=10^{1}\)
\(=10\)

\(4^{\frac{1}{3}} \times\left[2^{\frac{1}{3}} \times 3^{\frac{1}{2}}\right] \div 9^{\frac{1}{4}}\)
\(=\left(2^{2}\right)^{\frac{1}{3}} \times\left[2^{\frac{1}{3}} \times 3^{\frac{1}{2}}\right] \div\left(3^{2}\right)^{\frac{1}{4}}\)
\(=2^{\frac{2}{3}} \times 2^{\frac{1}{3}} \times 3^{\frac{1}{2}} \times \frac{1}{3^{\frac{1}{2}}}\)
\(=2^{\frac{2}{3}+\frac{1}{3}}\)
\(=2^{\frac{2+1}{3}}\)
\(=2^{\frac{3}{3}}\)
\(=2^{1}\)
\(=2\)

\(\left(8 a^{3} \div 27 x^{-3}\right)^{\frac{2}{3}} \times\left(64 a^{3} \div 27 x^{-3}\right)^{\frac{-2}{3}}\)
\(=\left[(2 \mathrm{a})^{3} \div\left(\frac{3}{x}\right)^{3}\right]^{\frac{2}{3}} \times\left[(4 \mathrm{a})^{3} \div\left(\frac{3}{x}\right)^{3}\right]^{-\frac{2}{3}}\)
\(=\left[(2 a)^{3} \times\left(\frac{x}{3}\right)^{3}\right]^{\frac{2}{3}} \times\left[(4 a)^{3} \times\left(\frac{x}{3}\right)^{3}\right]^{-\frac{2}{3}}\)
\(=\left[\left(2 a \times \frac{x}{3}\right)^{3}\right]^{\frac{2}{3}} \times\left[\left(4 a \times \frac{x}{3}\right)^{3}\right]^{-\frac{2}{3}}\)
\( =\left(\frac{2 a x}{3}\right)^{2} \times\left(\frac{4 a x}{3}\right)^{-2}\)
\(=\left(\frac{2 a x}{3}\right)^{2} \times\left(\frac{3}{4 a x}\right)^{2} \)
\(=\left(\frac{2 a x}{3} \times \frac{3}{4 a x}\right)^{2}\)
\(=\left(\frac{1}{2}\right)^{2}\)
\(=\frac{1}{4}\)

\(\left\{\left(x^{-5}\right)^{\frac{2}{3}}\right\}^{\frac{-3}{10}}\)
\(=\left\{x^{\frac{-10}{3}}\right\}^{-\frac{3}{10}}\quad\left[\because\left(a^{m}\right)^{n}=a^{m n}\right]\)
\(=x^{1}\)
\(=x\)

\(\left[\left\{\left(2^{-1}\right)^{-1}\right\}^{-1}\right]^{-1}\)
\(=\left[\left\{2^{-1}\right\}^{-1}\right]^{1}\)
\(=\left[2^{-1}\right]^{-1}\)
\(=2^{1}\)
\(=2\)

\(\sqrt[3]{a^{-2}} \cdot b \times \sqrt[3]{b^{-2}} \cdot c \times \sqrt[3]{c^{-2}} \cdot a\)
\( =a^{-\frac{2}{3}} \cdot b \times b^{-\frac{2}{3}} \cdot c \times c^{-\frac{2}{3}} \cdot a \)
\(=a^{1-\frac{2}{3}} \times b^{1-\frac{2}{3}} \times c^{1-\frac{2}{3}}\)
\(=a^{\frac{1}{3}} b^{\frac{1}{3}} c^{\frac{1}{3}} \)
\(=(a b c)^{\frac{1}{3}}\)
\(=\sqrt[3]{a b c} \)

\(\left(\frac{4^{m+\frac{1}{4}} \times \sqrt{2.2^{m}}}{2 . \sqrt{2^{-m}}}\right)^{\frac{1}{m}}\)
\(=\left[\frac{\left(2^{2}\right)^{m+\frac{1}{4}} \times \sqrt{2^{1+m}}}{2 \cdot 2^{-\frac{m}{2}}}\right]^{\frac{1}{m}} \)
\(=\left[\frac{2^{2 m+\frac{1}{2}} \times 2^{\frac{1+m}{2}}}{2^{1-\frac{m}{2}}}\right]^{\frac{1}{m}}\)
\(=\left[\frac{2^{2 m+\frac{1}{2}+\frac{1+m}{2}}}{2^{1-\frac{m}{2}}}\right]^{\frac{1}{m}} \)
\(=\left[2^{2 m+\frac{1}{2}+\frac{1}{2}+\frac{m}{2}-1+\frac{m}{2}}\right]^{\frac{1}{m}} \)
\(=\left[2^{2 m+\frac{m}{2}+\frac{m}{2}+\frac{1}{2}+\frac{1}{2}-1}\right]^{\frac{1}{m}}\)
\(=\left[2^{3 m+1-1}\right]^{\frac{1}{m}}\)
\(=\left(2^{3 m}\right)^{\frac{1}{m}}\)
\(=2^{3}\)
\(=8\)

\(9^{-3} \times \frac{16^{\frac{1}{4}}}{6^{-2}} \times\left(\frac{1}{27}\right)^{-\frac{4}{3}}\)
\( =\left(3^{2}\right)^{-3} \times \frac{\left(2 \right)^{\frac{1}{4}}}{(3.2)^{-2}} \times\left(\frac{1}{3}\right)^{3 \times\left(-\frac{4}{3}\right)}\)
\(=3^{-6} \times \frac{2}{3^{-2} \times 2^{-2}} \times\left(\frac{1}{3}\right)^{-4}\)
\(=3^{-6+2+4} \times 2^{1+2}\)
\(=3^{0} \times 2^{3}\)
\(=8\quad\left[\because 3^{0}=1\right] \)

\(\left(\frac{x^{a}}{x^{b}}\right)^{a^{2}+a b+b^{2}}\times\left(\frac{x^{b}}{x^{c}}\right)^{b^{2}+b c+c^{2}} \times\left(\frac{x^{c}}{x^{a}}\right)^{c^{2}+c a+a^{2}}\)
\(=x^{(a-b)}\left(a^{2}+a b+b^{2}\right) \times x^{(b-c)}\left(b^{2}+b c+c^{2}\right)\times x^{(c-a)\left(c^{2}+c a+a^{2}\right)}\)
\( =x^{a^{3}-b^{3}} \times x^{b^{3}-c^{3} }\times x^{c^{3}-a^{3}} \)
\(=x^{a^{3}-b^{3}+b^{3}-c^{3}+c^{3}-a^{3}}\)
\(=x^{0}\)
\(=1 \)

\(5^{\frac{1}{2}}, 10^{\frac{1}{4}}, 6^{\frac{1}{3}}\)
\(\because 2,4,3\)-এর ল.সা.গু \(12\)
\( 5^{\frac{1}{2}}=5^{\frac{6}{12}}=\sqrt[12]{5^{6}}=\sqrt[12]{15625} \)
\(10^{\frac{1}{4}}=10^{\frac{3}{12}}=\sqrt[12]{10^{3}}=\sqrt[12]{1000} \)
\(6^{\frac{1}{3}}=6^{\frac{4}{12}}=\sqrt[12]{6^{4}}=\sqrt[12]{1296} \)
\(\because \) \(1000 < 1296 < 15625 \)
\(\therefore\) \(\sqrt[12]{1000} < \sqrt[12]{1296} < \sqrt[12]{15625} \)
\(\therefore\) \(5^{\frac{1}{2}} > 6^{\frac{1}{3}} > 10^{\frac{1}{4}} \)

\(3^{\frac{1}{3}}, 2^{\frac{1}{2}}, 8^{\frac{1}{4}}\)
\(\because 2,4,3\)-এর ল.সা.গু \(12\)
\( 3^{\frac{1}{3}}=3^{\frac{4}{12}}=\sqrt[12]{3^{4}}=\sqrt[12]{81}\)
\(2^{\frac{1}{2}}=2^{\frac{6}{12}}=\sqrt[12]{2^{6}}=\sqrt[12]{64} \)
\(8^{\frac{1}{4}}=8^{\frac{3}{12}}=\sqrt[12]{8^{3}}=\sqrt[12]{512} \)
\(\because \) \(64 < 81 < 512 \)
\(\therefore\) \(\sqrt[12]{64} < \sqrt[12]{81} < \sqrt[12]{512} \)
\(\therefore\) \(2^{\frac{1}{2}} < 3^{\frac{1}{3}} < 8^{\frac{1}{4}} \)

\(2^{60}, 3^{48}, 4^{36}, 5^{24} \)
\(2^{60}=\left(2^{5}\right)^{12}=(32)^{12} \)
\(3^{48}=\left(3^{4}\right)^{12}=(81)^{12} \)
\(4^{36}=\left(4^{3}\right)^{12}=(64)^{12} \)
\(5^{24}=\left(5^{2}\right)^{12}=(25)^{12} \)
\(\because \) \(25 < 32 < 64 < 81 \)
\(\therefore\) \((25)^{12} < (32)^{12} < (64)^{12} < (81)^{12}\)
\(\therefore\) \(5^{24} < 2^{60} < 4^{36} < 3^{48} \)

বামপক্ষ, \(\left(\frac{a^{q}}{a^{r}}\right)^{p} \times\left(\frac{a^{r}}{a^{p}}\right)^{q} \times\left(\frac{a^{p}}{a^{q}}\right)^{r}\)
\(=a^{(q-r) p} \times a^{(r-p) q} \times a^{(p-q) r}\)
\(=a^{p q-p r+q r-p q+p r-q r}\)
\(=a^{0}=1=\) ডানপক্ষ (প্রমাণিত)

বামপক্ষ, \(\left(\frac{x^{\mathrm{m}}}{x^{\mathrm{n}}}\right)^{\mathrm{m}+\mathrm{n}}\left(\frac{x^{\mathrm{n}}}{x^{l}}\right)^{\mathrm{n}+l}\left(\frac{x^{l}}{x^{\mathrm{m}}}\right)^{l+\mathrm{m}}\)
\(=x^{(\mathrm{m}-\mathrm{n})(\mathrm{m}+\mathrm{n})} \times x^{(\mathrm{n}-l)(\mathrm{n}+l)} \times x^{(l-\mathrm{m})(l+\mathrm{m})}\)
\(=x^{m^{2}-n^{2}} \times x^{n^{2}-l^{2}} \times l^{2}-m^{2}\)
\(=x^{\mathrm{m}^{2}-\mathrm{n}^{2}+\mathrm{n}^{2}-l^{2}+l^{2}-\mathrm{m}^{2}}\)
\(=x^{0}=1=\) ডানপক্ষ (প্রমাণিত)

বামপক্ষ, \(\left(\frac{x^{\mathrm{m}}}{x^{\mathrm{n}}}\right)^{\mathrm{m}+\mathrm{n}-l} \times\left(\frac{x^{\mathrm{n}}}{x^{l}}\right)^{\mathrm{n}+l-\mathrm{m}} \times\left(\frac{x^{l}}{x^{\mathrm{m}}}\right)^{l+\mathrm{m}-\mathrm{n}}\)
\( =\left(x^{\mathrm{m}-\mathrm{n}}\right)^{\mathrm{m}+\mathrm{n}-l} \times\left(x^{\mathrm{n}-l}\right)^{\mathrm{n}+l-\mathrm{m}} \times\left(x^{l-\mathrm{m}}\right)^{l+\mathrm{m}-\mathrm{n}} \)
\( =x^{(\mathrm{m}-\mathrm{n})(\mathrm{m}+\mathrm{n})-l(\mathrm{m}-\mathrm{n})} \times x^{(\mathrm{n}-l)(\mathrm{n}+l)-\mathrm{m}(\mathrm{n}-l)} \times x^{(l-\mathrm{m})(l+\mathrm{m})-\mathrm{n}(l-\mathrm{m})} \)
\( = x^{m^{2}-n^{2}-l m+l n} \times x^{n^{2}-l^{2}-m n}+\mathrm{ml} \times x^{l^{2}}-\mathrm{m}^{2}-\mathrm{nl} +\mathrm{mn} \)
\(=x^{m^{2}-n^{2}-l m+ln+n^{2}-l^{2}-m n+m l+l^{2}-m l^{2}-n l+m n}\)
\(=x^{0}=1=\) ডানপক্ষ (প্রমাণিত)

বামপক্ষ, \(\left(a^{\frac{1}{x-y}}\right)^{\frac{1}{x-z}} \times\left(a^{\frac{1}{y-z}}\right)^{\frac{1}{y-x}} \times\left(a^{\frac{1}{z-x}}\right)^{\frac{1}{z-y}}\)
\( =a^{\frac{1}{(x-y)(x-z)}} \times a^{\frac{1}{(y-z)(y-x)}} \times a^{\frac{1}{(z-x)(z-y)}} \)
\(=a^{\frac{1}{(x-y)(x-z)}+\frac{1}{(y-z)(y-x)}+\frac{1}{(z-x)(z-y)}} \)
\(=a^{\frac{-1}{(x-y)(z-x)}+\frac{-1}{(y-z)(x-y)}+\frac{1}{(z-x)(y-z)}} \)
\( =a^{-\left[\frac{1}{(x-y)(z-x)}+\frac{1}{(y-z)(x-y)}+\frac{1}{(z-x)(y-z)}\right]} \)
\(=a^{-\left[\frac{y-z+z-x+x-y}{(x-y)(y-z)(z-x)}\right]}\)
\(=a^{-\frac{0}{(x-y)(y-z)(z-x)}} \)
\(=a^{0}=1=\) ডানপক্ষ (প্রমাণিত)

প্রদত্ত, \(x+z=2 y\)
বা, \(x-y=y-z\)
বামপক্ষ, \(a^{y-z} \cdot b^{z-x} \cdot c^{x-y}\)
\( =a^{x-y} \cdot c^{x-y} \cdot b^{z-x}\quad[\because x-y=y-z] \)
\(=(a c)^{x-y} \cdot b^{z-x} \)
\(=\left(b^{2}\right)^{x-y} \cdot b^{z-x} \quad\left[\because b^{2}=a c\right] \)
\(=b^{2 x-2 y+z-x}\)
\(=b^{x+z-2 y} \)
\(=b^{2 y-2 y}\quad[\because x+z=2 y] \)
\(=b^{0}=1=\) ডানপক্ষ (প্রমাণিত)

বামপক্ষ, \(a^{q-r} b^{r-p} c^{p-q}\)
\(=\left(x y^{p-1}\right)^{q-r}\left(x y^{q-1}\right)^{r-p}\left(x y^{r-1}\right)^{p-q}\quad{\left[\because a=x y^{p-1} b=x y^{q-1} c=x y^{r-1}\right]}\)
\(=x^{q-r} y^{(p-1)(q-r)} \cdot x^{r-p} y^{(q-1)(r-p)}.x^{p-q} y^{(r-1)(p-q)}\)
\(=x^{q-r+r-p+p-q} y^{(p-1)(q-r)+(q-1)(r-p)+(r-1)(p-q)}\)
\(=x^{0} \cdot y ^{p q-p r-q+r+q r-p q-r+p+p r-q r-p +q }\)
\(=1 \cdot y^{0}=1 \cdot 1=1=\) ডানপক্ষ (প্রমাণিত)

\(x^{\frac{1}{a}}=y^{\frac{1}{b}}=z^{\frac{1}{c}}=K\) (ধরি) \(\quad[\mathrm{K} \neq 0]\)
\(\therefore x^{\frac{1}{a}}=\mathrm{K}, y^{\frac{1}{b}}=\mathrm{K},\) এবং \(z^{\frac{1}{c}}=K\)
\(\therefore x=\mathrm{K}^{\mathrm{a}},\)
\( \therefore \mathrm{y}=\mathrm{K}^{\mathrm{b}}, \)
\(\therefore \mathrm{z}=\mathrm{K}^{\mathrm{c}}\)
প্রদত্ত, \(xyz=1\)
বা, \(\mathrm{K}^{a} \cdot \mathrm{K}^{b} \cdot \mathrm{K}^{c}=K^{0}\)
বা, \(\mathrm{K}^{\mathrm{a}+\mathrm{b}+\mathrm{c}}=\mathrm{K}^{0}\)
\(\therefore a+b+c=0\) (প্রমাণিত)

\(a^{x}=b^{y}=c^{z}=K\) (ধরি) \(\quad[\mathrm{K} \neq 0]\)
বা, \(a=K^{\frac{1}{x}}, b=K^{\frac{1}{y}}\) এবং \(c=K^{\frac{1}{z}}\)
প্রদত্ত, \(a b c=1\)
বা, \(K^{\frac{1}{x}} \cdot K^{\frac{1}{y}} \cdot K^{\frac{1}{z}}=1\)
বা, \(K^{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}}=K^{0}\)
বা, \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
বা, \(\frac{y z+x z+x y}{x y z}=0\)
বা, \(x y+y z+z x=0\) (প্রমাণিত)

\(49^{x}=7^{3}\)
বা, \(\left(7^{2}\right)^{x}=7^{3}\)
বা, \(7^{2 x}=7^{3} \)
\(\therefore 2 x=3\)
বা, \(x=\frac{3}{2}=1 \frac{1}{2} \)
\(\therefore\) নির্ণেয় সমাধান, \(x=1 \frac{1}{2}\)

\( 2^{x+2}+2^{x-1}=9 \)
বা, \(2^{x} \cdot 2^{2}+2^{x} \cdot 2^{-1}=9\)
বা, \(4 \cdot 2^{x}+\frac{2^{x}}{2}=9 \)
বা, \(2^{x} \cdot\left(4+\frac{1}{2}\right)=9 \)
বা, \(2^{x} \cdot \frac{9}{2}=9\)
বা, \(2^{x}=9 \times \frac{2}{9}\)
বা, \(2^{x}=2^{1} \)
বা, \(x=1 \)
\(\therefore\) নির্ণেয় সমাধান \(x=1\)

\( 2^{x+1}+2^{x+2}=48 \)
বা, \(2^{x} \cdot 2^{1}+2^{x} \cdot 2^{2}=48 \)
বা, \(2^{x}(2+4)=48 \)
বা, \(2^{x} \cdot 6=48\)
বা, \(2^{x}=\frac{48 }{6}\)
বা, \(2^{x}=2^{3}\)
বা, \(x=3\)
\(\therefore\) নির্ণেয় সমাধান \(x=3\)

\( 2^{4 x} \cdot 4^{3 x-1}=\frac{4^{2 x}}{2^{3 x}}\)
বা, \(2^{4 x} \cdot\left(2^{2}\right)^{3 x-1}=\frac{\left(2^{2}\right)^{2 x}}{2^{3 x}} \)
বা, \(2^{4 x} \cdot 2^{6 x-2}=2^{4 x-3 x} \)
বা, \(2^{4 x+6 x-2}=2^{4 x-3 x}\)
বা, \(2^{10 x-2}=2^{x} \)
\(\therefore\) \(10 x-2=x \)
বা, \(10 x-x=2 \)
বা, \(9 x=2\)
বা, \(x=\frac{2}{9} \)
\(\therefore\) নির্ণেয় সমাধান \(x=\frac{2}{9} \)

\(9 \times 81^{x}=27^{2-x}\)
বা, \(3^{2} \times\left(3^{4}\right)^{x}=\left(3^{3}\right)^{2-x} \)
বা, \(3^{2} \times 3^{4 x}=3^{6-3 x} \)
বা, \(3^{2+4 x}=3^{6-3 x}\)
\(\therefore\) \(2+4 x=6-3 x\)
বা, \(4 x+3 x=6-2\)
বা, \(7 x=4\)
\(\therefore\) \(x=\frac{4}{7} \)
\(\therefore\) নির্ণেয় সমাধান \(x=\frac{4}{7} \)

\(2^{5 x+4}+2^{9}=2^{10}\)
বা, \(2^{5 x+4}=2^{10}-2^{9}\)
বা, \(2^{5 x+4}=2^{9} \cdot(2-1)\)
বা, \(2^{5 x+4}=2^{9}\)
\(\therefore\) \(5 x+4=9\)
বা, \(5 x=9-4 \)
বা, \(5 x=5\)
বা, \(x=1 \)
\(\therefore\) নির্ণেয় সমাধান \(x=1 \)

\( 6^{2 x+4}=3^{3 x} \cdot 2^{x+8}\)
বা, \((2 \cdot 3)^{2 x+4}=3^{3 x} \cdot 2^{x+8}\)
বা, \(2^{2 x+4} \times 3^{2 x+4}=3^{3 x} \times 2^{x+8}\)
বা, \(\frac{2^{2 x+4}}{2^{x+8}}=\frac{3^{3 x}}{3^{2 x+4}}\)
বা, \(2^{2 x+4-x-8}=3^{3 x-2 x-4}\)
বা, \(2^{x-4}=3^{x-4}\)
বা, \(\left(\frac{2}{3}\right)^{x-4}=1\)
বা, \(\left(\frac{2}{3}\right)^{x-4}=\left(\frac{2}{3}\right)^{0}\)
\(\therefore\) \(x-4=0\)
\(\therefore\) \(x=4 \)
\(\therefore\) নির্ণেয় সমাধান \(x=4 \)

\((0.243)^{0.2} \times(10)^{0.6}\)
\(=\left(\frac{243}{1000}\right)^{\frac{2}{10}} \times(10)^{\frac{6}{10}}\)
\(=\left(\frac{3^{5}}{10^{3}}\right)^{\frac{1}{5}} \times(10)^{\frac{3}{5}}\)
\(=\frac{3^{5 \times \frac{1}{5}}}{10^{3 \times \frac{1}{5}}} \times 10^{\frac{3}{5}}\)
\(=\frac{3}{10^{\frac{3}{5}}} \times 10^{\frac{3}{5}}\)
\(=3\)

\(2^{\frac{1}{2}} \times 2^{-\frac{1}{2}} \times(16)^{\frac{1}{2}}\)
\( =2^{\frac{1}{2}} \times 2^{-\frac{1}{2}} \times 16^{\frac{1}{2}}\)
\(=2^{\frac{1}{2}+\left(-\frac{1}{2}\right)} \times\left(4^{2}\right)^{\frac{1}{2}}\)
\(=2^{0} \times 4\)
\(=1 \times 4\)
\(=4 \)

\(4^{x}=8^{3}\)
বা, \(\left(2^{2}\right)^{x}=\left(2^{3}\right)^{3}\)
বা, \( 2^{2 x}=2^{9} \)
বা, \(2 x=9\)
\(\therefore\) \(x=\frac{9}{2} \)

\(20^{-x}=\frac{1}{7}\)
বা, \(\left(20^{-x}\right)^{-2}=\left(\frac{1}{7}\right)^{-2}\)
বা, \(20^{2 x}=7^{2}\)
\(\therefore 20^{2 x}=49\)

\(4 \times 5^{x}=500\)
বা, \( 5^{x}=\frac{500 }{4}\)
বা, \(5^{x}=125\)
বা, \(5^{x}=5^{3}\)
\(\therefore\) \(x=3\)
\(\therefore\) \(x^{x}=3^{3}=27 \)

\((27^{x})=(81^{y})\)
বা, \(\left(3^{3}\right)^{x}=\left(3^{4}\right)^{y}\)
বা, \(3^{3 x}=3^{4 y}\)
\(\therefore 3 x=4 y\)
বা, \(\frac{x}{y}=\frac{4}{3} \)
বা, \(x: y=4: 3\)

\(\left(5^{5}+0.01\right)^{2}-\left(5^{5}-0.01\right)^{2}=5^{x}\)
বা, \( 4 \times 5^{5} \times 0.01=5^{x}\quad\left[\because(a+b)^{2}-(a-b)^{2}=4 a b\right]\)
বা, \(4 \times 5^{5} \times \frac{1}{100 }=5^{x}\)
বা, \(\frac{5^{5}}{5^{2}}=5^{x} \)
বা, \(5^{5-2}=5^{x}\)
বা, \( 5^{3}=5^{x} \)
\(\therefore\) \(x=3 \)

\( 3 \times 27^{x}=9^{x+4} \)
বা, \(3 \times\left(3^{3}\right)^{x}=\left(3^{2}\right)^{x+4} \)
বা, \(3 \times 3^{3 x}=3^{2 x+8}\)
বা, \(3^{1+3 x}=3^{2 x+8}\)
\(\therefore\) \(1+3 x=2 x+8\)
বা, \(3 x-2 x=8-1 \)
\(\therefore\) \(x=7 \)

\( \sqrt[3]{\left(\frac{1}{64}\right)^{\frac{1}{2}}}\)
\(=\sqrt[3]{\left(\frac{1}{64}\right)^{\frac{1}{2}}}\)
\(=\sqrt[3]{\left(\frac{1}{8}\right)^{2 \times \frac{1}{2}}}\)
\(=\sqrt[3]{\left(\frac{1}{8}\right)}\)
\(=\sqrt[3]{\left(\frac{1}{2}\right)^{3}}\)
\(=\sqrt[3]{2^{-3}}\)
\(=2^{-3 \times \frac{1}{3}}\)
\(=2^{-1}\)
\(=\frac{1}{2} \)

\( 3^{3^{3}}=3^{27}\)
\(\left(3^{3}\right)^{3}=3^{9}\)
\(\because \) \(27 > 9\)
\(\therefore\) \(3^{27} > 3^{9} \)
অর্থাৎ, \(3^{3^{3}} > \left(3^{3}\right)^{3} \)